We also show that a Banach space is Asplund if and only if every weak* compact subset has weak* slices whose index of non-separability is arbitrarily small. 1.
B U L L . AUSTRAL. M A T H . S O C .
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A CONTINUITY PROPERTY RELATED TO AN INDEX OF NON-SEPARABILITY AND ITS APPLICATIONS WARREN B . M O O R S
For a set E in a metric space X the index of non-separability is P(E) = inf{r > 0: E is covered by a countable-family of balls of radius less than r}. Now, for a set-valued mapping $ from a topological space A into subsets of a metric space X we say that $ is /3 upper semi-continuous at t 6 A if given e > 0 there exists a neighbourhood U of t such that /3($(f/)) < e. In this paper we show that if the subdifferential mapping of a continuous convex function $ is P upper semi-continuous on a dense subset of its domain then $ is Frechet differentiate on a dense Gs subset of its domain. We also show that a Banach space is Asplund if and only if every weak* compact subset has weak* slices whose index of non-separability is arbitrarily small. 1. INTRODUCTION
For a bounded set E in a metric space X the Kuratowski index of non-compactness of E is a.(E) = inf {r > 0: E is covered by a finite family of sets of diameter less than r}. Recently, in a paper by Giles and Moors [2], a new continuity property related to Kuratowski's index of non-compactness was examined. In that paper they said that a set-valued mapping $ from a topological space A into subsets of a metric space X is a upper semi-continuous at a point t G A if given e > 0 there exists an open neighbourhood U of t € A such that a($(U)) < e. They showed that under suitable circumstances a upper semi-continuity characterises (metric) upper semi-continuity, and that a significant class of set-valued mappings which are a upper semi-continuous on a dense subset of their domain are single-valued and (metric) upper semi-continuous on a dense Gg subset of their domain. In this paper we consider a natural generalisation of a upper semi-continuity called /? upper semi-continuity. This new upper semi-continuity condition is defined in terms of an index of non-separability. Received 9th July, 1991. Copyright Clearance Centre, Inc. Serial-fee code: 0004-9729/92
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[2]
In Section 2 we define the index of non-separability and prove some of its elementary properties which are analogous to those of the Kuratowski index of non-compactness. We also examine a property satisfied by the Kuratowski index of non-compactness, but which fails to be true for the index of non-separability. In Section 3 we show that any minimal weak* cusco from a Baire space A into subsets of the dual of a Banach space X which is /? upper semi-continuous on a dense subset of A, is single-valued and (norm) upper semi-continuous on a dense G& subset of A. In Section 4 we use the index of non-separability to define /3 denting points of a set in a Banach space and establish a pleasing connection between /? denting points and ordinary denting points. The single-valuedness property established in Section 3 suggests an application in determining conditions under which continuous convex functions on a Banach space are generically Frechet differentiable. In Section 5 we derive another characterisation of Asplund spaces. We also extend a recent result of Kenderov and Giles [3, Theorem 3.5] to show that on a Banach space X which can be equivalently renormed to have every point of the unit sphere of X a /? denting point of the closed unit ball of X, a continuous convex function on an open convex subset of X* is generically Frechet differentiable provided that the set of points where the function has a weak* continuous subgradient is residual in its domian. 2. A MEASURE OF NON-SEPARABILITY For a set E in a metric space X the index of non-separability is /3(E) = inf{r > 0: E is covered by a countable family of balls of radius less than r } , when E can be covered by a countable family of balls of fixed radii, otherwise, Throughout the rest of this paper all Banach spaces X are over the real numbers with dual X*. The closed unit ball {x E X: \\x\\ ^ 1} will be denoted by B(X) and the unit sphere {x E X: \\x\\ — 1} by S(X). Consider a non-empty bounded subset K of X. Given f e X* \ {0} and 6 > 0, the slice of K defined by / and 6 is the set S(K, f, 6) = {x E K: f{x) > s u p / ( i f ) — 6}. For a metric space (X, d), given xo E X and r > 0 we will denote by B(xo, r) the open ball {x E X: d(x, xo) < r} and by B[XQ, r] the closed ball {x £ X: d(x, xo) ^ r } . For any set E in a topological space X we will denote by C(E) the complement of E in X and E the closure of E in X. We will denote the interior of E by int E and the boundary of E by dE. PROPOSITION 2 . 1 . For a metric space {X, d), the index of 13 on X satisfies the following properties: 1.
0(A) ftO for any A C X;
non-sepaiabihty
[3]
A continuity property
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0( A) = 0 if and only if A is a. separable subset of X; 0{A) ^0{B) if ACBCX; 0{A) = 0(1) for any AC X;
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p( U > 0 = sxvp{0(An): n G N} wiere An C X for all n £ N;
6.
/?(4 HB)^ min{0(A), /3{B)} for A, B C X.
We omit the proofs of the properties (1) to (6) as they are straightforward. PROPOSITION 2 . 2 . For a normed linear space (X, \\-\\), separability /3 on X satisfies the following additional properties: 7. 8. 9.
the index of non-
0{A + B)^0(A) + 0(B) for A,BCX; 0(kA)= \k\0{A) for ACX andfcG E ; 0(coA) — f3(A) for AC. X where coA denotes the convex hull of A.
PROOF: The proofs of the properties (7) to (9) are straightforward, with the possible exception of (9), which we now prove. Clearly fi(A) ^ (3(coA) as A C coA, so it is sufficient to show f3(A) ^ 0(coA). Given e > 0, choose a sequence {z n }^=i and an r > 0 such that
B[xn, r) and 0{A) 0, consider Oe = (J{ °P e n se ts U in A: diam$(!7) ^ 2e}. Now Oe is open; we will show Oe is dense in A. From Lemma 5.5 there exists a dense Gg subset G\ of A such that at every point t 6 Gi the mapping p where p(t) = inf{||/|| : / 6
